Quantum Mechanics - Ajoy Ghatak¶

In [1]:
import sympy as sp
import numpy as np
import scipy.constants as const
from scipy.integrate import quad
import matplotlib.pyplot as plt
import seaborn as sns
import plotly.express as px

Dirac Delta Function¶

Function G(x): $$ R_{\sigma}(x) = \frac{1}{2\sigma} \quad; \quad a-\sigma<x<a+\sigma $$ $$ = 0 \quad ; \quad otherwise $$

In [2]:
def R_sigma(sigma, a, x):
    if a-sigma<x<a+sigma:
        return 1/(2*sigma)
    else:
        return 0
In [3]:
x = np.linspace(0, 1, 10000)
a = 0.5
sigmas = [1,0.5,1e-1,1e-2,1e-3]

for sigma in sigmas:
    R_sigma1 = [R_sigma(sigma, a, i) for i in x]
    plt.plot(x, np.array(R_sigma1), label=f'$\sigma$ = {sigma}', lw=0.7)
plt.legend()
plt.xlabel('x')
plt.ylabel(r'$R_{\sigma}(x)$')
plt.grid()
plt.show()
In [4]:
x1 = list(x)
x2 = x1*np.array(sigmas).size
x2 = np.array(x2)
sigmas1 = sigmas*x.size
sigmas2 = np.array(sigmas1)
sigmas2 = sigmas2.reshape(x.size, np.array(sigmas).size).T
sigmas2 = sigmas2.reshape(x.size*np.array(sigmas).size)
In [5]:
R_sigma2 = []
for sigma in sigmas:
    for i in x:
        rs2 = R_sigma(sigma, a, i)
        R_sigma2.append(rs2)
R_sigma2 = np.array(R_sigma2)
In [6]:
print(x2.shape, sigmas2.shape, R_sigma2.shape)

(50000,) (50000,) (50000,)
In [7]:
fig = px.line(x=x2, y=R_sigma2, color=sigmas2, title='R_sigma(x)',
                labels=dict(x='x',y='R_sigma(x)',color='sigma'))
fig.show()

Function T(x): $$ T_{\sigma}(x) = 1 \quad; \quad x>a+\sigma $$ $$ = \frac{1}{2\sigma}(x-(a-\sigma)) \,; \, a-\sigma<x<a+\sigma $$ $$ = 0 \quad ; \quad otherwise $$

Also; $$ \frac{dT_{\sigma}}{dx} = R_{\sigma}(x) $$

In [8]:
def T_sigma(sigma, a, x):
    if x>a+sigma:
        return 1
    elif a-sigma<=x<=a+sigma:
        return 1/(2*sigma) * (x-(a-sigma))
    else:
        return 0
In [9]:
x = np.linspace(0, 1, 10000)
a = 0.5
sigmas = [0.5,0.2,1e-1,1e-2,1e-3]

for sigma in sigmas:
    T_sigma1 = [T_sigma(sigma, a, i) for i in x]
    T_sigma1 = np.array(T_sigma1)
    plt.plot(x, T_sigma1, label=f'$\sigma$ = {sigma}', lw=0.7)
plt.legend()
plt.xlabel('x')
plt.ylabel(r'$T_{\sigma}(x)$')
plt.grid()
plt.show()
In [10]:
x = np.linspace(0, 1, 10000)
a = 0.5
sigmas = [0.5,0.2,1e-1,1e-2,1e-3]

for sigma in sigmas:
    T_sigma1 = [T_sigma(sigma, a, i) for i in x]
    T_sigma1 = np.array(T_sigma1)
    T_sigmap1 = np.gradient(T_sigma1, x)
    plt.plot(x, T_sigmap1, label=f'$\sigma$ = {sigma}', lw=0.7)
plt.legend()
plt.xlabel('x')
plt.ylabel(r'$\dfrac{dT_{\sigma}}{dx}$')
plt.grid()
plt.show()
In [11]:
x1 = list(x)
x2 = x1*np.array(sigmas).size
x2 = np.array(x2)
sigmas1 = sigmas*x.size
sigmas2 = np.array(sigmas1)
sigmas2 = sigmas2.reshape(x.size, np.array(sigmas).size).T
sigmas2 = sigmas2.reshape(x.size*np.array(sigmas).size)
In [12]:
T_sigma2 = []
for sigma in sigmas:
    for i in x:
        ts2 = T_sigma(sigma, a, i)
        T_sigma2.append(ts2)
T_sigma2 = np.array(T_sigma2)
In [13]:
print(x2.shape, sigmas2.shape, T_sigma2.shape)

(50000,) (50000,) (50000,)
In [14]:
fig = px.line(x=x2, y=T_sigma2, color=sigmas2, title='T_sigma(x)',
            labels=dict(x='x',y='T_sigma(x)',color='sigma'))
fig.show()
In [15]:
T_sigmap2 = np.gradient(T_sigma2, x2)

fig = px.line(x=x2, y=T_sigmap2, color=sigmas2, title='dT_sigma/dx',
            labels=dict(x='x',y='dT_sigma/dx',color='sigma'))
fig.show()

Gaussian Function: $$ G_{\sigma}(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) $$

In [16]:
def G_sigma(sigma, mu, x):
    return 1/(sigma*(2*np.pi)**0.5) * np.exp(-(x-mu)**2/(2*sigma**2))
In [17]:
x = np.linspace(0, 1, 10000)
a = 0.5
sigmas = [1,0.5,1e-1,1e-2,1e-3]

for sigma in sigmas:
    G_sigma1 = [G_sigma(sigma, a, i) for i in x]
    plt.plot(x, np.array(G_sigma1), label=f'$\sigma$ = {sigma}', lw=0.7)
plt.legend()
plt.xlabel('x')
plt.ylabel(r'$G_{\sigma}(x)$')
plt.grid()
plt.show()
In [18]:
x1 = list(x)
x2 = x1*np.array(sigmas).size
x2 = np.array(x2)
sigmas1 = sigmas*x.size
sigmas2 = np.array(sigmas1)
sigmas2 = sigmas2.reshape(x.size, np.array(sigmas).size).T
sigmas2 = sigmas2.reshape(x.size*np.array(sigmas).size)
In [19]:
G_sigma2 = []
for sigma in sigmas:
    for i in x:
        gs2 = G_sigma(sigma, a, i)
        G_sigma2.append(gs2)
G_sigma2 = np.array(G_sigma2)
In [20]:
print(x2.shape, sigmas2.shape, G_sigma2.shape)

(50000,) (50000,) (50000,)
In [21]:
fig = px.line(x=x2, y=G_sigma2, color=sigmas2, title=r'G_sigma(x)',
              labels=dict(x='x',y='G_sigma(x)',color='sigma'))
fig.show()

By Integration:

In [22]:
def dirac_delta_intg(k, a, x):
    integrand = 1/(2*np.pi) * np.exp(1j*k*(x-a))
    return quad(integrand, -np.inf, np.inf)
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